Proof theory is a branch of mathematical logic founded by David Hilbert around 1920 to pursue Hilbert’s programme. The problems addressed by the programme had already been formulated, in some sense, at the turn of the century, for example, in Hilbert’s famous address to the First International Congress of Mathematicians in Paris. They were closely connected to the set-theoretic foundations for analysis investigated by Cantor and Dedekind – in particular, to difficulties with the unrestricted notion of system or set; they were also related to the philosophical conflict with Kronecker on the very nature of mathematics. At that time, the central issue for Hilbert was the ‘consistency of sets’ in Cantor’s sense. Hilbert suggested that the existence of consistent sets, for example, the set of real numbers, could be secured by proving the consistency of a suitable, characterizing axiom system, but indicated only vaguely how to give such proofs model-theoretically. Four years later, Hilbert departed radically from these indications and proposed a novel way of attacking the consistency problem for theories. This approach required, first of all, a strict formalization of mathematics together with logic; then, the syntactic configurations of the joint formalism would be considered as mathematical objects; finally, mathematical arguments would be used to show that contradictory formulas cannot be derived by the logical rules.
This two-pronged approach of developing substantial parts of mathematics in formal theories (set theory, second-order arithmetic, finite type theory and still others) and of proving their consistency (or the consistency of significant sub-theories) was sharpened in lectures beginning in 1917 and then pursued systematically in the 1920s by Hilbert and a group of collaborators including Paul Bernays, Wilhelm Ackermann and John von Neumann. In particular, the formalizability of analysis in a second-order theory was verified by Hilbert in those very early lectures. So it was possible to focus on the second prong, namely to establish the consistency of ‘arithmetic’ (second-order number theory and set theory) by elementary mathematical, ‘finitist’ means. This part of the task proved to be much more recalcitrant than expected, and only limited results were obtained. That the limitation was inevitable was explained in 1931 by Gödel’s theorems; indeed, they refuted the attempt to establish consistency on a finitist basis – as soon as it was realized that finitist considerations could be carried out in a small fragment of first-order arithmetic. This led to the formulation of a general reductive programme.
Gentzen and Gödel made the first contributions to this programme by establishing the consistency of classical first-order arithmetic – Peano arithmetic (PA) – relative to intuitionistic arithmetic – Heyting arithmetic. In 1936 Gentzen proved the consistency of PA relative to a quantifier-free theory of arithmetic that included transfinite recursion up to the first epsilon number, ε0; in his 1941 Yale lectures, Gödel proved the consistency of the same theory relative to a theory of computable functionals of finite type. These two fundamental theorems turned out to be most important for subsequent proof-theoretic work. Currently it is known how to analyse, in Gentzen’s style, strong subsystems of second-order arithmetic and set theory. The first prong of proof-theoretic investigations, the actual formal development of parts of mathematics, has also been pursued – with a surprising result: the bulk of classical analysis can be developed in theories that are conservative over (fragments of) first-order arithmetic.
Countable sets versus sets that are countable in reverse mathematics